We study two McKean-Vlasov equations involving hitting times. Let $(B(t); t \geq 0)$ be standard Brownian motion, and $\tau:= \inf\{t \geq 0: X(t) \leq 0\}$ be the hitting time to zero of a given process $X$. The first equation is $X(t) = X(0) + B(t) - \alpha \mathbb{P}(\tau \leq t)$.

We provide a simple condition on $\alpha$ and the distribution of $X(0)$ such that the corresponding Fokker-Planck equation has no blow-up, and thus the McKean-Vlasov dynamics is well-defined for all time $t \geq 0$. We take the PDE approach and develop a new comparison principle.

The second equation is $X(t) = X(0) + \beta t + B(t) + \alpha \log \mathbb{P}(\tau \leq t)$, $t \geq 0$, whose Fokker-Planck equation is non-local. We prove that if $\beta,1/\alpha > 0$ are sufficiently large, the McKean-Vlasov dynamics is well-defined for all time $t \geq 0$. The argument is based on a relative entropy analysis. This is joint work with Erhan Bayraktar, Gaoyue Guo and Wenpin Tang.